Abstract:
Phase separation in composite materials, or materials that are made up of two or more
different components, is a process in which the material separates over time into regions of
predominantly one component or the other. Because these composite materials are often
developed to be used under certain conditions, significant changes of this nature in their
material composition can dramatically affect their performance, thus it is important to
understand the dynamic behavior of these changes.
One mathematical model of phase separation is the classical phase field model, which
is a system of nonlinear evolution equations that describes nonisothermal phase separation.
Instead of a composite material, the phase field model describes pure materials which can
assume two different phases; for example, a solid and a liquid phase.
In this dissertation, I analyze the behavior of solutions of the phase field model, along with two extensions to the classical model. One extension is the addition of stochastic
terms to the model, thus incorporating random behavior into the equations to address
fluctuations in temperature, structural imperfections of the material, or other unknown
effects. The other extension is the addition of a nonlocal operator to account for long range
interactions.
The behavior of all three variations of the phase field model is studied through numerical
simulations, and the results compared to describe the differences between the models. In
addition, I present a proof showing that the numerical method converges to the true solution
of the phase field model.
